I think you are mistating Kazdan-Warner's result. Class (C) consists of manifolds not in (A), (B). Not of manifolds that "only admit negative scalar curvature metrics".

There is a substantial literature on the subject. Let me focus on closed simply-connected manifolds of dimension at least $5$. With these assumptions

1. A manifold in the class (A) if and only if it is either non-spin, or it is spin and the Lichnerowicz-Hitchin obstruction vanishes.

2. A manifold is in class (B) if and only if its Lichnerowicz-Hitchin obstruction is nonzero, and it is the product of Ricci-flat Kaehler or Spin(7) manifolds. In this case the manifold is spin and even-dimensional. This is proved by A.Futaki in "Scalar-flat closed manifolds not admitting positive scalar curvature metrics" [Inventiones Math., 1993].

3. In any dimension $9+8k$ where $k$ is nonnegative integer there is a homotopy sphere with nonzero Lichnerowicz-Hitchin obstruction. Thus it lies in class (C). It cannot lie in (B) because it is odd-dimensional. I am sure there are many more examples.

There is a substantial literature on the subject. Let me focus on closed simply-connected manifolds of dimension at least $5$. With these assumptions

1. A manifold in the class (A) if and only if it is either non-spin, or it is spin and the Lichnerowicz-Hitchin obstruction vanishes.

2. A manifold is in class (B) if and only if its Lichnerowicz-Hitchin obstruction is nonzero, and it is the product of Ricci-flat Kaehler or Spin(7) manifolds. In this case the manifold is spin and even-dimensional. This is proved by A.Futaki in "Scalar-flat closed manifolds not admitting positive scalar curvature metrics" [Inventiones Math., 1993].

3. In any dimension $9+8k$ where $k$ is nonnegative integer there is a homotopy sphere with nonzero Lichnerowicz-Hitchin obstruction. Thus it lies in class (C). It cannot lie in (B) because it is odd-dimensional. I am sure there are many more examples.